For a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

We compute the total Ij -cohomology of a projective bundle over a smooth scheme.

In this note we recall the history of Serre's problem and its beautiful solution given by D. Quillen in 1976

The following are personal reminiscences of my research years in algebraic K-theory and cyclic homology during which Dan Quillen was everyday present in my professional life.

In the late sixties (of the twentieth century) the groups K0;K1;K2 were known and well-studied. The group K0 had been introduced by Alexander Grothendieck, then came K1 by Hyman Bass [2] (as a variation of the Whitehead group), permitting one to generalize the notion of determinant, and finally K2 by John Milnor [9] and Michel Kervaire. The big problem was: how about Kn? Having in mind topological K-theory and all the other generalized (co)homological theories, one was expecting higher K-groups which satisfy similar axioms, in particular the Mayer-Vietoris exact sequence. The discovery by Richard Swan of the existence of an obstruction for this property to hold shed some embarrassment. What kind of properties should we ask of Kn? There were various attempts, for instance by Max Karoubi and Orlando Villamayor [4]. And suddenly Dan Quillen came with a candidate sharing a lot of nice properties. He had even two different constructions of the same object: the “+” construction and the “Q” construction [14, 15]. Not only did he propose a candidate but he already got a computation: the higher K-theory of finite fields. This was a fantastic step forward and Hyman Bass organized a two week conference at the Battelle Institute in Seattle during the summer of 1972, which was attended by Bass, Borel, Husemoller, Karoubi, Priddy, Quillen, Segal, Stasheff, Tate, Waldhausen, Wall and sixty other mathematicians. The Proceedings appeared as Springer Lecture Notes 341, 342 and 343. I met Quillen for the first time on this occasion.

We survey Quillen's contributions to the area of cyclic homology, apart from his very first result in [19].

The excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 → S → P → Q → 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.